Divisor sum function: sigma(n)

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144, 72, 195, 74, 114, 124, 140, 96, 168, 80, 186, 121, 126, 84, 224, 108, 132, 120, 180, 90, 234, 112, 168, 128, 144, 120, 252, 98, 171, 156

OFFSET

1

COMMENTS

Sum of positive divisors of n.

PROPERTIES

Multiplicative with a(p^k) = (p^(k+1)-1)/(p-1).

FORMULAS

a(n) = Sum_{d|n} d

a(n) = Sum_{q=1..n} c_q(n) * floor(n/q), where c_q(n) is Ramanujan's sum function.

a(n) = Sum_{k=1..n} gcd(n, k) / phi(n / gcd(n, k))

PROGRAMS

Perl

use ntheory qw(:all);
sub a { divisor_sum(shift) }

PARI/GP

a(n) = sigma(n);

Sidef

func a(n) { sigma(n) }

SEE ALSO

• Euler's totient function: G1

• Wikipedia, Divisor function.

• Weisstein, Eric W. Divisor Function. From MathWorld--A Wolfram Web Resource.